Problem: $ C = \left[\begin{array}{rr}3 & 4 \\ 5 & 2 \\ 4 & 3\end{array}\right]$ $ E = \left[\begin{array}{rr}-2 & 0 \\ 2 & 0\end{array}\right]$ What is $ C E$ ?
Solution: Because $ C$ has dimensions $(3\times2)$ and $ E$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ C E = \left[\begin{array}{rr}{3} & {4} \\ {5} & {2} \\ \color{gray}{4} & \color{gray}{3}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{0} \\ {2} & \color{#DF0030}{0}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rr}{3}\cdot{-2}+{4}\cdot{2} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{-2}+{4}\cdot{2} & ? \\ {5}\cdot{-2}+{2}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{-2}+{4}\cdot{2} & {3}\cdot\color{#DF0030}{0}+{4}\cdot\color{#DF0030}{0} \\ {5}\cdot{-2}+{2}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{3}\cdot{-2}+{4}\cdot{2} & {3}\cdot\color{#DF0030}{0}+{4}\cdot\color{#DF0030}{0} \\ {5}\cdot{-2}+{2}\cdot{2} & {5}\cdot\color{#DF0030}{0}+{2}\cdot\color{#DF0030}{0} \\ \color{gray}{4}\cdot{-2}+\color{gray}{3}\cdot{2} & \color{gray}{4}\cdot\color{#DF0030}{0}+\color{gray}{3}\cdot\color{#DF0030}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}2 & 0 \\ -6 & 0 \\ -2 & 0\end{array}\right] $